已知正三角形ABC边长为a,用这个三角形的高为边,作一个新的正三角形,再用这第二个正三角形的高为边作正三角形,…,这样无限继续下去,则所有正三角形的面积之和为 .
【答案】
分析:先设第n个三角形的面积为a
n,根据三角形面积公式得出a
1,a
2,a
3,发现数列{a
n}为等比数列,进而求出前n项和的极限,即可得到答案.
解答:解:设第n个三角形的面积为a
n,则a
1=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/0.png)
×a×a×sin60°=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/1.png)
a
2,
a
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/2.png)
×
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/3.png)
a×
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/4.png)
a×sin60°=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/5.png)
a
2=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/6.png)
×
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/7.png)
a
2,
a
3=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/8.png)
×
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/9.png)
a×
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/10.png)
a×sin60°=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/11.png)
×
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/12.png)
a
2;
…
∴数列{a
n}为首项为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/13.png)
a
2,公比为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/14.png)
的等比数列.
所有这些三角形的面积的和为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/15.png)
(a
1+a
2+…+a
n)=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/16.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/17.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/18.png)
a
2.
故答案为:
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024183211351199632/SYS201310241832113511996010_DA/19.png)
a
2.
点评:本题主要考查了等比数列的应用以及相似三角形的性质,相似三角形面积的比等于相似比的平方.